Nearest just interval
An irrational interval or ratio of frequencies given by a real number r has an infinite list of nearest just intervals; if r is rational, the list is finite, terminating in r. For arbitrary (including negative) real numbers this corresponds to what number theorists call best rational approximations. A ratio of integers p/q with q > 0 and p and q relatively prime is a best rational approximation if there is no ratio m/n with n < q which is a better approximation to r. If r is an interval of music it is positive, and both p and q are positive. Note that a nearest just interval is not necessarily nearest in logarithmic terms; 4/3 and 3/2 are the same distance in cents from √2 = 600 cents, but |4/3 - √2| = .08088 whereas |3/2 - √2| = 0.08479, which is larger.
Best rational approximations also arise in music theory logarithmically, as the best rational approximations to the logarithm base two of some number such as 3/2 or ∜5 is often of interest.
The semiconvergents of the continued fraction for r include all of the best rational approximations. The convergents are equivalent with a stronger notion of best approximation, namely best relative approximation. Here it is required that |qr - p| is less than |nr - m| for any n < q.
Examples
Approximations for Ratios (of Pure Intervals)
The best rational approximations to log2(3/2) define edos which have especially good approximations to the fifth (701.955000865... cents):
Step\EDO | log(Tenney Height) | size in cents | "error" in cents |
... | ... | ... | ... |
1 \ 1 | 0.0 | 1200.0 | 498.04 |
1 \ 2 | 1.0 | 600.00 | -101.96 |
2 \ 3 | 2.585 | 800.00 | 98.045 |
3 \ 5 | 3.907 | 720.00 | 18.045 |
4 \ 7 | 4.807 | 685.7143 | -16.2407 |
7 \ 12 | 6.392 | 700.00 | -1.955 |
17 \ 29 | 8.945 | 703.4483 | 1.4933 |
24 \ 41 | 9.943 | 702.43902 | 0.48402 |
31 \ 53 | 10.682 | 701.88679 | -0.06821 |
- for approximations of the harmonic seventh see 7_4
Approximation for Logarihmic Measures
The 600-cent interval sqrt(2) (6 steps of 12edo, "Tritone") approximates following ratios:
freq. ratio | log2(Tenney Height) | size in cents | "error" in cents |
... | ... | ... | ... |
1 / 1 | 0.0 | 0.0 | 600.0 |
3 / 2 | 2.585 | 701.96 | 101.96 |
4 / 3 | 3.585 | 498.04 | -101.96 |
7 / 5 | 5.129 | 582.51 | -17.49 |
17 / 12 | 7.672 | 603.000 | 3.000 |
24 / 17 | 597.000 | -3.000 | |
99 / 70 | 600.0883 | 0.0883 | |
140 / 99 | 599.9117 | -0.0883 | |
... | ... | ... | ... |
The 300-cent interval 2^(1/4) (3 steps of 12edo, "minor third") approximates following ratios:
freq. ratio | log(Tenney Height) | size in cents | "error" in cents |
... | ... | ... | ... |
1 / 1 | 0.0 | 0.0 | 300.0 |
6 / 5 | 4.907 | 315.64 | 15.64 |
13 / 11 | 7.160 | 289.21 | -10.79 |
19 / 16 | 8.248 | 297.51 | -2.49 |
25 / 21 | 9.036 | 301.84 | 1.84 |
... | ... | ... | ... |